Orthogonal polynomials and diffusion operators

D. Bakry 1 , S. Orevkov 1 , M. Zani 2

”Orthogonal polynomials and Hypergroups” colloquium

Universite Paul Sabatier, June 19th 2014

1Institut de Mathematiques de Toulouse, Universite Paul Sabatier2Laboratoire d’Analyse et Mathematiques appliquees , Universite Paris-Est -Creteil

Question: when can we describe eigenvectors and eigenvalues of an operator?

Possible answer: when eigenvectors are orthogonal polynomials

Introduction

Consider a Markov diffusion process (Xt) with continuous trajectoriesIts law is characterized by Markov kernels

Pt(f )(x) = E(f (Xt)/X0 = x) , x x ∈ Rd ,

The infinitesimal generator L associated to (Pt)t≥0:

Lf = limt→∞

Pt f − f

t,

Denote F (x , t) = Pt(f )(x), F solution of the heat equation

∂tF = LF , F (x , 0) = f (x).

Introduction

L is a second order differential operator: (no zero order component)

L(f ) =∑

ij

g ij (x)∂2ij f +

∑i

bi (x)∂i f .

Ask L to be self-adjoint with respect to some measure µ, and the spectrum tobe discrete (µ is then said to be the reversible measure for (Xt)).When µ has a density ρ which is C1 with respect to the Lebesgue measure, andthe coefficients g ij assumed to be C1,

L(f ) =1

ρ

∑ij

∂i (g ijρ∂j f ).

We restrict our attention to operators which are elliptic in the interior of thesupport of µ.

Introduction

If there is an orthonormal basis (en) of L2(µ) composed of eigenvectors of L,

Len = −λnen,

then one has

Pt(f )(x) =

∫f (y)pt(x , y)dµ(y),

wherept(x , y) =

∑n

e−λnten(x)en(y) ,

and for fixed x , the function pt(x , y) is the density with respect to µ(dy) of thelaw of Xt when X0 = x .

Introduction

Square field operator

Γ(f , g) =∑

ij

g ij∂i f ∂j g =1

2

(L(fg)− f L(g)− gL(f )

),

Change of variable: Φ : Rk 7→ R fi : Ω 7→ R smooth,

L(Φ(f1, · · · , fk )

)=∑

ij

∂2ij Φ(f)Γ(fi , fj ) +

∑i

∂i Φ(f)L(fi ).

Dimension 1

Classification:

On R : Hermite polynomials : µ(dx) = e−x2/2 dx√2π

.

On [0,∞) : Laguerre polynomials : µ(dx) = Caxae−x dx .

On [−1, 1] : Jacobi polynomials µ(dx) = Ca,b(1− x)a(1 + x)bdx .

In those three examples, the associated polynomials are eigenvectors ofDiffusion Operators

• Hermite case : L(f ) = f ′′ − xf ′, LPn = −nPn. O-U process

• Laguerre case : L(f ) = xf ′′ − (a + 1− x)f ′, L(Pn) = −nPn squared radialO-U process

• Jacobi case : L(f ) = (1− x2)f ′′ − ((a− b) + (a + b − 2)x)f ′,L(Pn) = −n(n + a + b − 1)Pn. Jacobi process

Geometric interpretation for Jacobi

Laplace operator on spheres Sn−1 ⊂ Rn.L(xi ) = −(n − 1)xi .Γ(xi , xj ) = δij − xi xj .X := 2(x2

1 + · · · x2p )− 1, p ≤ n.

L(X ) = −2(n + 1)X + 2p, Γ(X ,X ) = 4(1− X 2).14L(Φ(X )) = L(Φ)(X )

4L(Φ)(X ) = Γ(X ,X )Φ′′(X ) + L(X )Φ′(X ).

L : Jacobi operator with parameters a = (n − p)/2 + 1, b = p/2 + 1.

Geometric interpretation for Hermite and Laguerre

Jacobi to Hermite scale Jacobi on (−√

n,√

n), a = b = n, n→∞Jacobi to Laguerre move and scale Jacobi on (0,

√n), limit a = n→∞, b

fixed.Hermite to Laguerre Hermite on Rd , applied on f (X ) with X = x2

1 + · · · x2d :

Laguerre with parameter a = d/2 + 1.

General setting

Ω be some open set in Rd , with piecewise C1 boundary.Let L be a diffusion operator

L(f ) =∑

ij

g ij (x)∂2ij f +

∑i

bi (x)∂i f ,

where g ij and bi are smooth functions on Ω, and the matrix (g ij ) is symmetric,positive definite for any x ∈ Ω.Let µ(dx) = ρ(x)dx be a probability measure with smooth density ρ on Ω forwhich polynomials are dense in L2(µ)Question: when does exist an orthonormal basis (Pn) of polynomials in L2(µ)which are eigenvectors for L?

LPn = −λnPn

Such eigenvalues (λn) turn out to be necessarily non negative.

General setting

RemarkIf Pd

n denotes the vector space of polynomials in d variables and total degreesmaller than n, and Hd

n the space of polynomials with total degree n,orthogonal to Pd

n−1 in Pdn , then

dim Pdn =

(n + d

n

), and dim Hd

n =

(n + d − 1

n

).

The choice of an orthonormal basis made of polynomials in L2(µ) amounts tothe choice of a basis for Hd

n , for any n.

General setting

RemarkThe restriction of L to Pd

n being symmetric for any n, one has, for any pair(P,Q) of polynomials ∫

Ω

PL(Q)dµ =

∫QL(P)dµ.

the restriction of L to polynomials is entirely determined by Γ (hence by thematrices (g ij (x))x∈Ω), and the measure µ

SDOP Problem

In Rd

Definition (DOP problem)

Let Ω be an open set with piecewise smooth boundary (which may be empty),µ(dx) = ρ(x)dx a probability measure with smooth positive density on Ω, suchthat polynomials are dense in L2(µ), and let L an elliptic diffusion operatorwith smooth coefficients on Ω. We say that (Ω,L, µ) is a solution to theDiffusion-Orthogonal Polynomials problem (in short DOP problem) if thereexists a complete basis of L2(µ) formed with orthogonal polynomials which areat the same time eigenvectors for the operator L.

Definition (SDOP problem)

Strong Diffusion Orthogonal Polynomial problem = DOP problem + for any fand h smooth and compactly supported in Rd∫

Ω

f L(h) dµ =

∫Ω

hL(f ) dµ.

SDOP Problem

In Rd

Writing µ(dx) = ρ(x) dx , for any f smooth and compactly supported in Ω,L(f ) may be defined by formula

L(f ) =1

ρ

∑ij

∂i

(g ijρ∂j f

),

Hence L is entirely determined from the (co)metric g = (g ij ) and the measuredensity ρ(x). We therefore talk about the triple (Ω, g , ρ) as a solution of theSDOP problem.

SDOP Problem

In Rd

Proposition

If L is a solution to the DOP problem, then bi (x) ∈ Pd1 and for any

i , j = 1, · · · , d, g ij (x) ∈ Pd2 .

Let ∆1, · · · ,∆p be the real irreducible components of ∆ = det(g ij ). Since noone of them may vanish in Ω, we always assume that they are positive in Ω.

TheoremSuppose that (Ω, g , ρ) is a solution to the SDOP problem. Then ∂Ω lies in∆ = 0. Moreover, if ∆1 · · ·∆r denotes the reduced equation of ∂Ω, then forany k = 1, · · · , r , and any i = 1, · · · , d, there exists some polynomial S i

k ∈ Pd1

such that, for any i = 1, · · · , d∑j

g ij∂j ∆k = S ik ∆k .

SDOP Problem

In Rd

Theorem (Conversely)

1- Conversely, assume that Ω is relatively compact and has a boundary ∂Ωwhich is an algebraic hyper-surface with reduced equation∆ = ∆1 · · ·∆r = 0. Suppose moreover that there exists a (co)metric(g ij )(x) positive definite on Ω such that for any (i , j), g ij ∈ Pd

2 and, forany k, for any i = 1, · · · , d, there exists some S i

k ∈ Pk1 such that∑

j

g ij∂j ∆k = S ik ∆k . (1)

Then, for any real numbers a1, · · · , ar for which ρ(x) = ∆a11 · · ·∆

arr is

integrable on Ω, (Ω, g ,Cρ) is a solution to the DOP problem, where C isthe normalizing constant.

2- The conclusion of point 1 still holds if one replaces the family of equations(1), ∑

ij

g ij∂j ∆ = S∆,

for some S ∈ Pd1 and where ∆ is the reduced equation of ∂Ω.

Idea of the proof

∫Ω

f Lhdµ =

∫Ω

hLfdµ

for any smooth compactly supported f and h imposes∑j

g ij nj = 0

on the boundary (n = unit normal vector)Then, (g) degenerates at the boundary. det(g) = 0 on ∂Ω.If ∂Ω ⊂ Q = 0, then Q is factor of det(g), and on Q = 0,

g ij∂j Q = 0,

Concludes by Hilbert’s Nullstellensatz

Form of the measure

∆ ∈ Pd2d , and we decompose it into irreducible real factors.

∆ = ∆m11 · · ·∆

mpp .

Furthermore, for every irreducible real factor ∆j which may be factorized inC[X ,Y ], we write this factorization as ∆j = (Rj + iIj )(Rj − iIj ). Let J theset of indices i ∈ 1, · · · , p such that ∆i is complex reducible.

Proposition (General form of the measure)

Then, there exist real constants (αi , βj ), and some polynomial Q withdeg(Q) ≤ 2n − deg(∆), such that

ρ =∏

i

|∆i |αi exp( Q

∆m1−11 · · ·∆mp−1

p

+∑j∈J

βj arctanIj

Rj

).

Result in R2

TheoremIn R2, up to affine transformations, there are exactly a one parameter familyand 10 relatively compact sets for which there exist a solution for the DOPproblem

Algebraic formulation in natural degree

Let

g =

(a bb c

)be a symmetric matrix whose coefficients a, b, c are polynomials of degree 2 inR[x , y ]. Let ∆ a square free polynomial in R[x , y ] which is factor of∆ := ac − b2 such that for each irreducible factor ∆1 of ∆,

a∂1∆1 + b∂2∆1 = L1∆1

b∂1∆1 + c∂2∆1 = L2∆1

where deg Li ≤ 1, i = 1, 2. We want to describe such a, b, c, ∆.

The 11 compact models in dimension 2 : triangle

Figure : Triangle

Equation : xy(1− x − y) = 0.Measure ρ(x) = xay b(1− x − y)c .

The 11 compact models in dimension 2 : circle

Figure : Circle

Equation : (1− x2 − y 2) = 0.Measure ρ(x) = (1− x2 − y 2)a.

The 11 compact models in dimension 2 : square

Figure : Square

Equation : (1− x)(1 + x)(1− y)(1 + y) = 0.Measure ρ(x) = (1− x)a(1 + x)b(1− y)c (1 + y)d .

The 11 compact models in dimension 2 : double parabola

Figure : Coaxial Parabolas

Equation : (y − x2 + 1)(y − 1 + αx2) = 0.Measure ρ(x) = (y − x2 + 1)a(y − 1 + αx2)b.

The 11 compact models in dimension 2 : Parabola with two lines 1

Figure : Parabola with two lines 1

Equation : (y − x2)y(1− x) = 0.Measure ρ(x) = (y − x2)ay b(1− x)c .

The 11 compact models in dimension 2 : Parabola with two lines 2

Figure : Parabola with two lines 2

Equation : (y − x2)(y + 2x + 1)(y − 2x + 1) = 0.Measure ρ(x) = (y − x2)a(y + 2x + 1)b(y − 2x + 1)c .

The 11 compact models in dimension 2 : Cuspidal Cubic 1

Figure : Cuspidal cubic 1

Equation : (y 2 − x3)(1− x) = 0.Measure ρ(x) = (y 2 − x3)a(1− x)b.

The 11 compact models in dimension 2 : Cuspidal Cubic 2

Figure : Cuspidal cubic 2

Equation : (y 2 − x3)(2y − 3x + 2) = 0.Measure ρ(x) = (y 2 − x3)a(2y − 3x + 2)b.

The 11 compact models in dimension 2 : Nodal Cubic

Figure : Nodal Cubic

Equation : y 2 − x2(1− x) = 0.Measure ρ(x) = (y 2 − x2(1− x))a.

The 11 compact models in dimension 2 : Swallow Tail

Figure : Swallow Tail

Equation :4 x2 − 27 x4 + 16 y − 128 y 2 − 144 x2y + 256 y 3 = 0Measure ρ(x) = (4 x2 − 27 x4 + 16 y − 128 y 2 − 144 x2y + 256 y 3)a.

The 11 compact models in dimension 2 : Deltoid

Figure : Deltoid

Equation : (x2 + y 2)2 + 18(x2 + y 2)− 8x3 + 24xy 2 − 27 = 0.

Measure ρ(x) =(

(x2 + y 2)2 + 18(x2 + y 2)− 8x3 + 24xy 2 − 27)a

.

Remarks on the models

• When the degree of ∂Ω is 4, this boundary is ∆ = 0 Hence anadmissible measure is ρ(x) = ∆−1/2, which corresponds to theLaplace-Beltrami operator associated with the (co-)metric g [operator hasconstant curvature, either 0 or positive]

• Interpretation as Laplace–Beltrami operators ∆G acting on(X ,Y ) : G → R2

”Dictionary” linking the angles of the reflection associated to thesymmetries and the type of singularities of the boundary of Ω:· double points → π/2· cusps → π/3· double tangents → π/4

Remarks on the models

1- Square: Product of Jacobi

2- Circle : the metric is not uniquea=b=0 : constant curvatureIf p = −1/2 operator is Laplace operator on S2, acting on functions whichare invariant under the symmetry x3 7→ −x3

If p = (d − 3)/2 we consider functions depending only on (x1, x2)L image of ∆Sd through the projection x 7→ (x1, x2)

3- Triangle: the metric not uniqueTake the circle model acting on x2 = X and y 2 = Y → triangle model

Remarks on the models

4- Coaxial parabolas:One parameter family Y = X 2 − 1 and Y = 1− aX 2 (a > −1)

Ga =

(1− 1

2(1 + a)X 2 X ((1− a)− (1 + a)Y )

X ((1− a)− (1 + a)Y ) 2(1 + a)(1− Y 2)− 4aX 2

)If a 6= 0, the degree is 4, ∆−1/2 admissible measure.If a = 0 : parabolic biangle (see T. Koornwinder)

5- Parabola + tangent and secant : degree 4, p = −1/2 comes fromLaplacian on the sphere S2

Remarks on the models

6- Parabola with two tangents:Constant curvature 0Take density measure (X 2 − Y )−1/2(Y − 2X + 1)−1/2(Y + 2X + 1)−1/2:L image of a 2-dimensional Euclidean Laplacian, constructed from the rootsystem B2

Roots λj = ±√

2ei (canonic basis (e1, e2))X (x , y) = 1

4

∑4j=1 exp(iλj .(x , y)) = (cos(

√2x) + cos(

√2y))/2

Y (x , y) = 14

∑4j=1 exp(iµj .(x , y)) = cos(

√2x) cos(

√2y).

7- Nodal cubic : this model has degree 3Unique metric up to scaling

G =

(4X (1− X ) 2Y (2− 3X )

2Y (2− 3X ) 4X − 3X 2 − 9Y 2

).

The boundary has degree 3, the measure density ρ(x) = det(G)−1/2 is notan admissible measureTake ρp(X ,Y ) =

(X 2(1− X )− Y 2

)p, p = −1/2, comes from Laplace

operator on S3

Remarks on the models

8- Cuspidal cubic + secantGeneral measure

ρp1,p2 = (1− X )p1 (X 3 − Y 2)p2

p1 = p2 = −1/2: image of ∆S2 through

X = (x1)2 + (x2)2, Y = x1((x1)2 − 3(x2)2).Functions invariant under symmetries through H = x3 = 0 and the twohyperplanes having an angle ±π/3 with H

Remarks on the models

9- Cuspidal cubic with one tangent: degree 4, comes from sphere

10- Swallow tail : curvature is 2

11- Deltoid: Curvature 0Comes from Euclidean Laplacian in dim 2

Z = e2i(1·z) + e2i(j·z) + e2i (j·z),

L−1/2 is the image of ∆R2 through ZZ is invariant under the symmetries w.r.t.

D1 = =(z) = 0, D2 = te iπ/3, t ∈ R, D3 = ae iπ/6 + te2iπ/3,

(a = π/√

3)→ root system A2 (see T. Koornwinder)

Last remark

∆SO(d)(x ij ) = −(d − 1)x ij

ΓSO(d)(xkl , xpq) = δkpδlq − xkqxpl .

∆SU(d)(z ij ) = −2(d − 1)(d + 1)

dz ij ,

ΓSU(d)(zkl , zpq) = −2zkqzpl +2

dzkl zpq

ΓSU(d)(zkl , zpq) = 2(δkpδlq − 1

dzkl zpq).

Deltoid case: L1/2 is the image of 3∆SU(3) through Z

T. Koornwinder,Orthogonal polynomials in two variables which are eigenfunctions of twoalgebraically independent partial differential operators. i., ii, iii, iv, Nederl. Akad.Wetensch. Proc. Ser. A 77=Indag. Math. 36 (1974).

T. Koornwinder,Special functions associated with root systems: recent progress, From universalmorphisms to megabytes: a Baayen space odyssey, Math. Centrum, CentrumWisk. Inform., Amsterdam,, 1994

T. Koornwinder and A. L. Schwartz,Product formulas and associated hypergroups for orthogonal polynomials on thesimplex and on a parabolic biangle, Constr. Approx. 13 (1997)

H.L. Krall and I.M. Sheffer,

Orthogonal polynomials in two variables, Ann. Mat. Pura Appl. 76 (1967)

C. Dunkl and Y. Xu.Orthogonal polynomials of several variables. In Encyclopedia of Mathematics andits Applications, volume 81. Cambridge University Press, Cambridge, 2001.

L I. G. Macdonald.Affine Hecke algebras and orthogonal polynomials, volume 157 of CambridgeTracts in Mathematics. Cambridge University Press, Cambridge, 2003.

O. Mazet.Classification des semi–groupes de diffusion sur R associes a une famille depolynomes orthogonaux. In J. Azema et al, ed., Seminaire de probabilites XXXILecture Notes in Mathematics, volume 1655, Springer–Verlag, 1997.